Abstract
In a commutative ring R with unity, a unit u is called exceptional if \(u-1\) is also a unit. For \(R = {\mathbb {Z}}/n{\mathbb {Z}}\) and for any \(f(X) \in {\mathbb {Z}}[X]\), an element \({\overline{u}} \in {\mathbb {Z}}/n{\mathbb {Z}}\) is called an “f-exunit” if \(gcd(f(u),n) = 1\). Recently, we obtained the number of representations of a non-zero element of \({\mathbb {Z}}/n{\mathbb {Z}}\) as a sum of two f-exunits for a particular infinite family of polynomials \(f(X) \in {\mathbb {Z}}[X]\). In this paper, we complete this problem by proving a similar formula for any non-constant polynomial \(f(X) \in {\mathbb {Z}}[X]\).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Change history
03 April 2021
A Correction to this paper has been published: https://doi.org/10.1007/s00013-021-01586-0
References
Anand, Chattopadhyay, J., Roy, B.: On sums of polynomial-type exceptional units in \({\mathbb{Z} }/n{\mathbb{Z}}\). Arch. Math. (Basel) 114, 271–283 (2020)
Dolžan, D.: The sums of exceptional units in a finite ring. Arch. Math. (Basel) 112, 581–586 (2019)
Hu, S., Sha, M.: On the additive and multiplicative structures of the exceptional units in finite commutative rings. Publ. Math. Debrecen 94, 369–380 (2019)
Houriet, J.: Exceptional units and Euclidean number fields. Arch. Math. (Basel) 88, 425–433 (2007)
Lenstra, H.W.: Euclidean number fields of large degree. Invent. Math. 38, 237–254 (1977)
Louboutin, S.R.: Non-Galois cubic number fields with exceptional units. Publ. Math. Debrecen 91, 153–170 (2017)
Miguel, C.: On the sumsets of exceptional units in a finite commutative ring. Monatsh. Math. 186, 315–320 (2018)
Miguel, C.: Sums of exceptional units in finite commutative rings. Acta Arith. 188, 317–324 (2019)
Nagell, T.: Sur un type particulier d’unités algébriques. Ark. Mat. 8, 163–184 (1969)
Niklasch, G.: Counting exceptional units. Collect. Math. 48(1–2), 195–207 (1997). (Journées Arithmétiques, Barcelona (1995))
Niklasch, G., Smart, N.P.: Exceptional units in a family of quartic number fields. Math. Comp. 67, 759–772 (1998)
Poulakis, D.: Integer points on algebraic curves with exceptional units. J. Aust. Math. Soc. Ser. A 63, 145–164 (1997)
Sander, J.W.: On the addition of units and nonunits mod \(m\). J. Number Theory 129, 2260–2266 (2009)
Sander, J.W.: Sums of exceptional units in residue class rings. J. Number Theory 159, 1–6 (2016)
Silverman, J.H.: Exceptional units and numbers of small Mahler measure. Exp. Math. 4, 69–83 (1995)
Silverman, J.H.: Small Salem numbers, exceptional units, and Lehmer’s conjecture. Rocky Mountain J. Math. 26, 1099–1114 (1996)
Stewart, C.L.: Exceptional units and cyclic resultants. Acta Arith. 155, 407–418 (2012)
Yang, Q.H., Zhao, Q.Q.: On the sumsets of exceptional units in \({\mathbb{Z}}_n\). Monatsh. Math. 182, 489–493 (2017)
Acknowledgements
It is a pleasure to thank Prof. R. Thangadurai for his constant encouragement and support throughout the project. We are grateful to him for clearing up many doubts. We thank the anonymous referee for his/her useful comments that improved the exposition to a great extent. This work was completed when the first author was visiting Harish-Chandra Research Institute and he gratefully acknowledges the hospitality provided by the Institute.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
The original online version of this article was revised: in order to indicate the correct corresponding author.
Rights and permissions
About this article
Cite this article
Anand, Chattopadhyay, J. & Roy, B. On a question of f-exunits in \(\mathbb {Z}/n\mathbb {Z}\). Arch. Math. 116, 403–409 (2021). https://doi.org/10.1007/s00013-020-01558-w
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00013-020-01558-w